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**Using Assessment to Improve and Evaluate Student Learning in Introductory Statistics**

Bob delMas (Univ. of MN)

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**GAISE Guidelines Assessment needs to be aligned with learning goals**

Focus on learning key ideas (not only skills, procedures, and computed answers) Include formative assessments as well as summative Use timely feedback to promote learning It is possible to implement good assessments even in large classroom settings

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**Suggestions for Teachers**

Integrate assessment as an essential component of the course (timing of assessment and activities). Use a variety of assessment methods to provide a more complete evaluation of student learning. Assess statistical literacy using assessments such as interpreting or critiquing articles in the news and graphs in media. Assess statistical thinking using assessments such as student projects and open-ended investigative tasks.

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**Suggestions for Large Classes**

Use small group projects instead of individual projects. Use peer review of projects to provide feedback and improve projects before grading. Use items that focus on choosing good interpretations of graphs or selecting appropriate statistical procedures. Use discussion sections for student presentations.

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**AAHE 9 Principles of Assessment**

Assessment begins with educational values. Treats understanding of learning as multidimensional, integrated, and revealed over time. Requires clear, explicitly stated purposes. Requires attention to experiences that lead to outcomes. Works best when ongoing, not episodic. Fosters wider improvement when representatives from across the educational community are involved (all stakeholders). Represents issues and questions that people really care about. Most likely to lead to improvement if part of a larger set of conditions that promote change. Meets responsibilities to students and to the public.

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**National Research Council (2001), Knowing what Students Know**

Assessment Triangle National Research Council (2001), Knowing what Students Know Cognition: the aspects of achievement or competencies that are to be assessed Observation: the tasks used to collect evidence about students’ achievement (i.e., the assessments) Interpretation: the methods used to analyze the evidence resulting from the tasks The three elements are interdependent A successful assessment synchronizes all three elements

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**COGNITION A theory or set of beliefs about:**

How students represent knowledge How students develop competence Used to identify important knowledge and skills Based on a learning model that provides a level of detail sufficient to accomplish the assessment

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**OBSERVATION What we typically consider to be the “assessment”**

Careful design of tasks that will provide evidence that can be linked to the learning model Context and Purpose are Important A national assessment can indicate relative standing, but not sensitive to nuances of instruction Instructor constructed assessment can be tied to classroom instruction, but may not generalize to a larger population

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INTERPRETATION Set of assumptions and models that are used to interpret evidence from observation Links observations to competencies (cognition) National assessments may use formal, statistical models that identify patterns indicative of competency levels Classroom assessment is typically more qualitative and identifies categories of competency based on observations Important to identify what has not developed as well as what has developed Important to identify misunderstandings and misconceptions as well as correct understanding

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**Example: Understanding p-Values**

A p-value is the probability of obtaining results as or more extreme than the observed results, given that the null hypothesis is true. A p-value IS NOT the probability that the null hypothesis is true. A p-value IS NOT the probability that the alternative hypothesis is true.

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**Example: Understanding p-Values**

A research article reports the results of a new drug test. The drug is to be used to decrease vision loss in people with Macular Degeneration. The article gives a p-value of .04 in the analysis section. Indicate if the following interpretation of this p-value is valid or invalid. Statement (N = 1617) Valid Invalid The probability of getting results as extreme or more extreme than the ones in this study if the drug is actually not effective.

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**Example: Understanding p-Values**

A research article reports the results of a new drug test. The drug is to be used to decrease vision loss in people with Macular Degeneration. The article gives a p-value of .04 in the analysis section. Indicate if the following interpretation of this p-value is valid or invalid. Statement (N = 1617) Valid Invalid The probability of getting results as extreme or more extreme than the ones in this study if the drug is actually not effective. 57.7 42.3

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**Example: Understanding p-Values**

A research article reports the results of a new drug test. The drug is to be used to decrease vision loss in people with Macular Degeneration. The article gives a p-value of .04 in the analysis section. Items 25, 26, and 27 present three different interpretations of this p-value. Indicate if each interpretation is valid or invalid. Statement (N = 1617) Valid Invalid 25. The probability of getting results as extreme or more extreme than the ones in this study if the drug is actually not effective. 57.7 42.3 26. The probability that the drug is not effective. 27. The probability that the drug is effective.

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**Example: Understanding p-Values**

A research article reports the results of a new drug test. The drug is to be used to decrease vision loss in people with Macular Degeneration. The article gives a p-value of .04 in the analysis section. Items 25, 26, and 27 present three different interpretations of this p-value. Indicate if each interpretation is valid or invalid. Statement (N = 1617) Valid Invalid 25. The probability of getting results as extreme or more extreme than the ones in this study if the drug is actually not effective. 57.7 42.3 26. The probability that the drug is not effective. 39.9 60.1 27. The probability that the drug is effective. 45.2 54.8 9% of all students made an incorrect choice for all 3 items Of those who chose “valid” for Item 25 (N = 933) 55% chose valid for at least one of the other items 7% chose “valid” for all 3 items.

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**Assessment Cycle (from Beth Chance ; see G. Wiggins 1992, 1998)**

Set goals What should students know, be able to do? At what point in the course? Identify assessable learning outcomes that match goals Select methods Identify an assessment that matches the type of learning outcome Consider minute papers, article reviews, newspaper assignments, projects, short answer items, multiple choice Can the assessment be built into the activity? Gather evidence (i.e., administer the assessment) Draw inference Don’t use results just to assign a grade Consider what responses indicate about student understanding Take action Provide feedback What can be done to remedy a misunderstanding (an activity; extra reading; more experience with a procedure or a concept) Re-examine goals and methods

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**Embedding Assessment into Classroom Activities**

Sorting Distributions Goal: Learn to associate labels with shapes of distributions Normal Distribution Goal: Learn to find areas for the standard normal distribution Sampling Distributions Learn the characteristics of sampling distributions Understand effect of sample size

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**ARTIST Website https://app.gen.umn.edu/artist**

Item Database (Assessment Builder): A collection of about 1100 items, in a variety of item formats, organized according to statistical topic and type of learning outcome assessed. Resources: Information, guidelines, and examples of alternative assessments. Copies of articles or direct links to articles on assessment in statistics. References and links for other related assessment resources. Research Instruments: Instruments that can be used for research and evaluation projects that involve assessments of outcomes related to teaching and learning statistics. Implementation issues: Questions and answers on practical issues related to designing, administering, and evaluating assessments. Presentations: Copies of conference papers and presentations on the ARTIST project, and handouts from ARTIST workshops. Events: Information on ARTIST events. Participation: Ways to participate as a class tester for ARTIST materials.

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ARTIST Topic Tests There are 11 scales, consisting of 8-12 multiple-choice items, that can be administered online. Our goal is to develop high quality, valid and reliable scales that can be used for a variety of purposes (e.g., research, evaluation, review, or self-assessment). TOPICS Data Collection (data types, types of study, study design) Data Representation (choose appropriate graphs, interpret graphs) Measures of Center (estimate, when to use, interpret, properties) Measures of Spread (estimate, when to use, interpret, properties) Normal Distribution (characteristics, empirical rule, areas under the curve) Probability (interpret, independence, relative frequency, simulation) Bivariate Quantitative Data (scatterplots, correlation, descriptive and inferential methods, outliers, diagnostics, influential observations) Bivariate Categorical Data (two-way tables and chi-square test, association) Sampling Distributions (types of samples, sample variability, sampling distributions, Central Limit Theorem) Confidence Intervals (interpret, confidence level, standard error, margin of error) Tests of Significance (hypothesis statements, p-values, Type I and II error, statistical and practical significance)

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**Comprehensive Assessment of Outcomes in Statistics (CAOS)**

Forty item test that can be administered as an online test to evaluate the attainment of desired student outcomes. CAOS items are designed to represent the big ideas and the types of reasoning, thinking and literacy skills deemed important for all students across first courses in statistics. Unifying focus is on reasoning about variability: in univariate and bivariate distributions, in comparing groups, in samples, and when making estimates and inferences. Not intended to be used exclusively as a final exam or as the sole assessment to assign student grades. CAOS can provide very informative feedback to instructors about what students have learned and not learned in an introductory statistics course (e.g., administered as pretest and posttest).

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**Data Representation Item (page 19)**

A baseball fan likes to keep track of statistics for the local high school baseball team. One of the statistics she recorded is the proportion of hits obtained by each player based on the number of times at bat as shown in the table below. Which of the following graphs gives the best display of the distribution of proportion of hits in that it allows the baseball fan to describe the shape, center and spread of the variable, proportion of hits?

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**Data Representation Item (page 19)**

A baseball fan likes to keep track of statistics for the local high school baseball team. One of the statistics she recorded is the proportion of hits obtained by each player based on the number of times at bat as shown in the table below. Which of the following graphs gives the best display of the distribution of proportion of hits in that it allows the baseball fan to describe the shape, center and spread of the variable, proportion of hits? What percents would you predict for your students? RESPONSE PERCENT (N = 1643) Graph A Graph B Graph C Graph D

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**Data Representation Item (page 19)**

A baseball fan likes to keep track of statistics for the local high school baseball team. One of the statistics she recorded is the proportion of hits obtained by each player based on the number of times at bat as shown in the table below. Which of the following graphs gives the best display of the distribution of proportion of hits in that it allows the baseball fan to describe the shape, center and spread of the variable, proportion of hits? RESULTS US undergraduates RESPONSE PERCENT (N = 1643) Graph A 11.1 Graph B 46.4 Graph C 29.1 Graph D 13.4

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**Data Representation Item (page 20)**

A local running club has its own track and keeps accurate records of each member's individual best lap time around the track, so members can make comparisons with their peers. Here are graphs of these data. Which of the graphs allows you to most easily see the shape of the distribution of running times? RESPONSE PERCENT (N = 1345) Graph A Graph B Graph C All of the above

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**Data Representation Item (page 20)**

A local running club has its own track and keeps accurate records of each member's individual best lap time around the track, so members can make comparisons with their peers. Here are graphs of these data. Which of the above graphs allows you to most easily see the shape of the distribution of running times? RESPONSE PERCENT (N = 1345) Graph A 43.8 Graph B 48.9 Graph C 3.6 All of the above 3.7

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**First Small Group Exercise**

Designate one person to be the recorder. Discuss the following questions (pages of handout) with respect to the Data Representation items: Why do you think students are selecting the incorrect responses for these items? (3-5 minutes) Outline an instructional activity to help students develop the correct understanding. (10 minutes)

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**Second Small Group Exercise**

Choose a Topic: Sampling Variability Confidence Intervals Tests of Significance Bivariate Quantitative Data Discuss the following questions: Why do you think students are selecting the incorrect responses for each item? (5-10 minutes) Outline an instructional activity to help students develop the correct understanding. (10-15 minutes)

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**Sampling Variability Item 1**

A certain manufacturer claims that they produce 50% brown candies. Sam plans to buy a large family size bag of these candies and Kerry plans to buy a small fun size bag. Which bag is more likely to have more than 70% brown candies? RESPONSE PERCENT (N = 1608) Sam, because there are more candies, so his bag can have more brown candies. 5.3 Sam, because there is more variability in the proportion of browns among larger samples. 11.6 Kerry, because there is more variability in the proportion of browns among smaller samples. 32.4 Kerry, because most small bags will have more than 50% brown candies. 1.7 Both have the same chance because they are both random samples. 48.9

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**Sampling Variability Item 2**

Consider the distribution of average number of hours that college students spend sleeping each weeknight. This distribution is very skewed to the right, with a mean of 5 and a standard deviation of 1. A researcher plans to take a simple random sample of 18 college students. If we were to imagine that we could take all possible random samples of size 18 from the population of college students, the sampling distribution of average number of hours spent sleeping will have a shape that is RESPONSE PERCENT (N = 872) Exactly normal. 18.8 Less skewed than the population. 34.4 Just like the population (i.e., very skewed to the right). 34.7 It's impossible to predict the shape of the sampling distribution. 12.0

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**Confidence Interval Item 1**

Suppose two researchers want to estimate the proportion of American college students who favor abolishing the penny. They both want to have about the same margin of error to estimate this proportion. However, Researcher 1 wants to estimate with 99% confidence and Researcher 2 wants to estimate with 95% confidence. Which researcher would need more students for her study in order to obtain the desired margin of error? RESPONSE PERCENT (N = 1296) Researcher 1. 51.9 Researcher 2. 25.9 Both researchers would need the same number of subjects. 9.1 It is impossible to obtain the same margin of error with the two different confidence levels. 13.1

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**Confidence Interval Item 2**

A high school statistics class wants to estimate the average number of chocolate chips per cookie in a generic brand of chocolate chip cookies. They collect a random sample of cookies, count the chips in each cookie, and calculate a confidence interval for the average number of chips per cookie (18.6 to 21.3). Indicate if the following interpretations are valid or invalid. Statement (N = 1609) Valid Invalid We are 95% certain that each cookie for this brand has approximately 18.6 to 21.3 chocolate chips. 51.2 48.8 We expect 95% of the cookies to have between 18.6 and chocolate chips. 34.1 65.9 We would expect about 95% of all possible sample means from this population to be between 18.6 and chocolate chips. 53.1 46.9 We are 95% certain that the confidence interval of 18.6 to includes the true average number of chocolate chips per cookie. 75.7 24.3

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**Test of Significance Item 1**

A newspaper article claims that the average age for people who receive food stamps is 40 years. You believe that the average age is less than that. You take a random sample of 100 people who receive food stamps, and find their average age to be 39.2 years. You find that this is significantly lower than the age of 40 stated in the article (p < .05). What would be an appropriate interpretation of this result? RESPONSE PERCENT (N = 1101) The statistically significant result indicates that the majority of people who receive food stamps is younger than 40. 33.8 Although the result is statistically significant, the difference in age is not of practical importance. 50.5 An error must have been made. This difference is too small to be statistically significant. 15.7

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**Test of Significance Item 2**

A researcher compares men and women on 100 different variables using a two-sample t-test. He sets the level of significance to .05 and then carries out 100 independent t-tests (one for each variable) on data from the same sample. If, in each case, the null hypothesis actually is true for every test, about how many "statistically significant" findings will this researcher report? RESPONSE PERCENT (N = 1160) 30.2 5 45.7 10 7.1 None of the above 17.1

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**Bivariate Quantitative Data Item 1**

The number of people living on American farms has declined steadily during the last century. Data gathered on the U.S. farm population (millions of people) from 1910 to 2000 were used to generate the following regression equation: Predicted Farm Population = (YEAR). What method would you use to predict the number of people living on farms in 2050. RESPONSE PERCENT (N = 1591) Substitute the value of 2050 for YEAR in the regression equation, and compute the predicted farm population. 19.8 Plot the regression line on a scatterplot, locate 2050 on the horizontal axis, and read off the corresponding value of population on the vertical axis. 15.6 Neither method is appropriate for making a prediction for the year 2050 based on these data. 28.4 Both methods are appropriate for making a prediction for the year 2050 based on these data. 36.2

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**Bivariate Quantitative Data Item 2**

A statistics instructor wants to use the number of hours studied to predict exam scores in his class. He wants to use a linear regression model. Data from previous years shows that the average number of hours studying for a final exam in statistics is 8.5, with a standard deviation of 1.5, and the average exam score is 75, with a standard deviation of 15. The correlation is Should the instructor use linear regression to predict exam scores from hours studied? RESPONSE PERCENT (N = 850) Yes, there is a high correlation, so it is alright to use linear regression. 21.2 Yes, because linear regression is the statistical method used to make predictions when you have bivariate quantitative data. 27.1 Linear regression could be appropriate if the scatterplot shows a clear linear relationship. 46.2 No, because there is no way to prove that more hours of study causes higher exam scores. 5.5

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**Assessment Builder: Search**

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**Assessment Builder: Results**

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**Assessment Builder: Results**

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**Assessment Builder: Item Set**

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**Assessment Builder: Download**

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**Assessment Builder: Download**

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**ARTIST Website https://app.gen.umn.edu/artist**

We invite you to contact the ARTIST team with any comments and suggestions you have regarding this presentation, or any of the materials at the ARTIST website. Thank you for your participation in today’s session.

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